A Generalized Schawlow-Townes Limit from Heisenberg Uncertainty and Causality*

The Schawlow-Townes limit has long been known to impose a limit to the frequency stability of lasers and masers due to spontaneous emission in their gain media. We show that the Schawlow-Townes limit applies to a much broader class of devices called feedback oscillators, realized by a phase-insensitive amplifier in positive feedback. A generalized version of the Schawlow-Townes limit arises naturally in feedback oscillators due to the basic demands of Heisenberg uncertainty and causality. This generalized Schawlow-Townes limit holds for both "good cavity" and "bad cavity" oscillators, in which the linewidth limit may be set by the bandwidth of the device's feedback element or its amplifier. Recently realized bad-cavity oscillators such as super-radiant lasers and solid-state masers can saturate this generalized Schawlow-Townes limit. It can be surpassed through appropriate quantum engineering: for example by atomic spin squeezing in a super-radiant laser.